Theorem imass1 6060

Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)

Assertion

Ref	Expression
imass1	⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶))

Proof of Theorem imass1

Step	Hyp	Ref	Expression
1		ssres 5962	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶))
2		rnss 5888	. . 3 ⊢ ((𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶) → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶))
3	1, 2	syl 17	. 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶))
4		df-ima 5637	. 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
5		df-ima 5637	. 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
6	3, 4, 5	3sstr4g 3976	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3890  ran crn 5625   ↾ cres 5626   “ cima 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  predrelss  6295  vdwnnlem1  16957  dprdres  19996  imasnopn  23665  imasncld  23666  imasncls  23667  utoptop  24209  restutop  24212  ustuqtop3  24218  utopreg  24227  metustbl  24541  imadifxp  32686  gsumfs2d  33137  esum2d  34253  eulerpartlemmf  34535  bj-imdirco  37520  brtrclfv2  44172  frege97d  44197  frege109d  44202  frege131d  44209  hess  44225  resimass  45687  setrecsss  50188